�

λ-Calculus: Then & Now

Dana S. Scott

University Professor Emeritus Carnegie Mellon University

Visiting Scholar University of California, Berkeley

dana.scott@cs.cmu.edu

Taken from talks prepared for:

TURING CENTENNIAL CELEBRATION Princeton University, May 10-12, 2012

ACM TURING CENTENARY CELEBRATION San Francisco, June 15-16, 2012

�1

�

For a General History: F. Cardone and J.R. Hindley. “Lambda-Calculus and Combinators in the 20th Century.”

In: Volume 5, pp. 723-818, of Handbook of the History of Logic, Dov M. Gabbay and John Woods eds., North-Holland/Elsevier Science, 2009

Symbols of Princeton

�2

Traditional From the Graduate Alumni (to encourage ecology)

The λ-calculus was begun by A. Church at Princeton, but it has been recycled

every decade after the 1930s in new and useful ways.

���

���

�

�

�

�

�

�

What is the λ-Calculus?The calculus gives rules for the explicit definition of functions; however, this type-free version also

permits recursion and self-replication.

α-conversion λX.[...X...] = λY.[...Y...]

β-conversion(λX.[...X...])(T) = [...T...]

η-conversion λX.F(X) = F

�3

The names of the rules are due to H.B. Curry. The last rule fails in many interpretations, and special efforts are needed to make it valid.

Church’s original system (1932) also had rules for logic, but that was the system his students Kleene & Rosser

proved inconsistent (1936). Church also proposed his calculus as giving a definition of computability. But

K. Gödel objected and later preferred Turing’s definition.

�4

Church vs. Turing

Alonzo Church Born: 14 June 1903 in Washington, D.C., USA. Died: 11 Aug 1995 in Hudson, Ohio, USA. Ph.D.: Princeton University, 1927, USA

Alan Turing Born: 23 June 1912, Maida Vale, London, UK. Died: 7 June 1954, Wilmslow, Cheshire, UK. Ph.D.: Princeton University, 1938, USA.

Alonzo Church, “An Unsolvable Problem in Elementary Number Theory,” American J. of Mathematics, vol. 5 (1936), pp. 345-363.

Alonzo Church, “A Note on the Entscheidungsproblem,” J. of Symbolic Logic, vol. 1 (1936) pp. 40-41. Correction: ibid, pp. 101-102.

Alan Turing,“On Computable Numbers with an Application to the Entscheidungsproblem,” Proc. of the London Math. Soc., vol. 42 (1936), pp. 230-267. Correction: vol. 43 (1937), pp. 544-546.

Alan Turing,“Computability and λ-definability,” J. Symbolic Logic, vol. 2 (1937), pp. 153-163.

The work of Church and Turing in 1936 was done independently.

�5

Haskell Brooks Curry

Born: 12 Sept 1900 in Millis, MA, USA. Died: 1 Sept 1982 in State College, PA, USA. Ph.D.: Göttingen Universität, 1930, Germany. Thesis: Grundlagen der kombinatorischen Logik

Stephen Cole Kleene

Born: 5 Jan 1909 in Hartford, CN, USA. Died: 25 Jan 1994 in Madison, WI, USA. Ph.D.: Princeton University, 1934, USA. Thesis: A Theory of Positive Integers in Formal Logic

Born: 6 Dec 1907 in Jacksonville, FL, USA. Died: 5 Sept 1989 in Madison, WI, USA. Ph.D.: Princeton University, 1934, USA. Thesis: A Mathematical Logic without Variables

J. Barkley Rosser

It seems, sadly, that Alan Turing never had a chance to meet these people or Kurt Gödel.

Three Other Pioneers

�6

The Connection to Computability

0 = λF.λX.Xn+1 = λF.λX.F(n(F)(X))n+m = λF.λX.n(F)(m(F)(X))n×m = λF.n(m(F)) mn = n(m)

n-1 = [a little harder]

Kleene and Turing independently proved this in different ways.

Theorem. For every partial recursive function g(n), there is a constant λ-term G such that

G(n) = g(n) , for all n.

Y = λF.(λX.F(X(X)))(λX.F(X(X)))Y(F) = F(Y(F))

Fixed-Point Combinator

Church Numerals

�7

Some λ-Definitions

test = λN.λU.λV.snd(N(shft(λX.X))(pair(V)(U)))

mult = λN.λM.λF.N(M(F)) fact = λN.test(N)(1)(mult(N)(fact(pred(N))))

fact = Y(λF.λN.test(N)(1)(mult(N)(F(pred(N)))))

pair = λX.λY.λF. F(X)(Y)

fst = λP.P(λX.λY. X)

snd = λP.P(λX.λY. Y)

succ = λN.λF.λX.F(N(F)(X)) shft = λS.λP. pair(S(fst(P)))(fst(P))

pred = λN. snd(N(shft(succ))(pair(0)(0)))

Kleene’s “trick” here is to introduce pairs as a data

structure, and then apply iteration to get

a sequence of pairs.

The factorial function must be the most overdefined function in the history of mankind!

�8

Church-Turing Thesis accepted with the help of Kleene

after Turing explained his machines.

Effectively computable functions of natural numbers can be identified with

those definable by:

• λ-calculus • Herbrand-Gödel equations • Partial-recursive schemata

• Turing-Post machine programs

If Gödel had stayed in Princeton, and

If Church and Kleene had argued better for data structures in the λ-calculus,

Then surely Gödel would have accepted λ-calculus as a foundation much earlier.

Note that Kleene proved the equivalence with

Herbrand-Gödel computability before Turing’s work.

�9

Kleene’s Complaint

I myself, perhaps unduly influenced by rather chilly receptions from audiences around 1933-35 to disquisitions on λ-definability, chose, after general recursiveness had appeared, to put my work in that format. I did later publish one paper 1962 on λ-definability in higher recursion theory.

I thought general recursiveness came the closest to traditional mathematics. It spoke in a language familiar to mathematicians, extending the theory of special recursiveness, which derived from formulations of Dedekind and Peano in the mainstream of mathematics.

I cannot complain about my audiences after 1935, although whether the improvement came from switching I do not know. In retrospect, I now feel it was too bad I did not keep active in λ-definability as well. So I am glad that interest in λ-definability has revived, as illustrated by Dana Scott’s 1963 communication.

Were the truth to be known, Kleene translated much of what he had done in λ-calculus into working with

integers. Indeed, the application operation {e}(n)

defines a partial combinatory algebra with many properties similar to the work of Curry and Rosser.

�10

Does λ-Calculus have Models?

(n,m) = 2n(2m+1)

set(0) = ∅

set((n,m)) = set(n) ∪ { m }

X* = { n | set(n) ⊆ X }

Application

F(X) = { m | ∃n ∈ X*.(n,m) ∈

F }

Abstraction λX.[...X...] =

{0}∪{ (n,m) | m ∈ [... set(n)...] }

Every set of integers can be used as an enumeration operator. The operator is computable

if the set is r.e. Many compound contexts do define enumeration operators.

Yes! There is a calculus for enumeration operators! First we need some simple definitions on

integers and sets of integers:

�11

Turing’s Only Student

Born: 23 September 1919, Peppard, Oxon., UK. Died: 20 November 1995, Oxford, UK. Ph.D.: Cambridge, 1953. Thesis: On axiomatic systems in Mathematics and theories in Physics. Supervisor: Alan Turing. Reader: Oxford University, Wolfson College, 1969-1986. Students: 26 and 126 descendants.

Robin Oliver Gandy

It is interesting to note that both the teams of

Myhill and Shepherdson and, later,

Friedberg and Rogers

defined enumeration operators without seeing they

had models for the λ-calculus.

Another pioneer, Gandy, later became a key contributor

to the development of Recursive Function Theory.

�12

Church’s Solution

Theorem. Only a finite number of axioms are needed to define a non-recursive set of integers.

R.M.Robinson’s Arithmetic(1) ∀ x ∀ y [ x = y ⟺ Sx = Sy ]

(2) ∀ x [ x = 0 ⟺ ¬∃y. x = Sy ]

(3) ∀ x ∀ y [ (x + 0) = x & (x + Sy) = S(x + y) ]

(4) ∀ x ∀ y [ (x × 0) = 0 & (x × Sy) = ((x × y) + x) ]

After the solution of Hilbert’s 10th Problem, the applicability of this theory

became even easier.

What is the Entscheidungsproblem?

To determine whether a formula of the first-order predicate calculus

is provable or not.

�13

Turing’s SolutionTheorem. Only a finite number of axioms are

needed to define the Universal Turing Machine.

Minskyizing the UTSStarting with Claude Shannon in 1956, many people —

often in competition with Marvin Minsky — proposed very small UTMs (but their operation requires extensive coding

of patterns). But, axiomatically, they do not require as many axioms as Turing did.

Post-Markov’s SolutionThe basic idea of Post (1943) was that a logistic system

is simply a set of rules specifying how to change one string of symbols (antecedent) into another string

of symbols (consequent). This leads to:

The Word Problem for Semigroups(1) ∀ x ∀ y [ x 1 = x = 1 x ]

(2) ∀ x ∀ y ∀ z [ x (y z) = (x y) z ]

Problem: Determine the provability of

A0 = B0 & A1 = B1 & ... & An-1 = Bn-1 ⟹ An = Bn .

�14

Schönfinkel-Curry’s SolutionSchönfinkel in 1924 and then Curry in 1929, both at

Göttingen, began the study of combinators, which were quickly connected with Church’s λ-calculus of 1932.

From them — with hindsight — we get:

Another Undecidable Theory(1) ∀ x ∀ y [ K(x)(y) = x ]

(2) ∀ x ∀ y ∀ z [ S(x)(y)(z) = x(z)(y(z)) ]

(3) ¬ K = S

Problem: Determine the provability of T = 0.

The only problem with this theory is that you either need models or something like the

Church-Rosser Theorem to know it is consistent. A weaker theory of

deterministic reduction can be given a fairly short

axiomatization and then be proved consistent

by much simpler means.

�15

McCarthy, LISP, & λ-CalculusLISP History according to McCarthy's memory in 1978. Presented at the ACM SIGPLAN History of Programming Languages Conference, June 1-3, 1978. It was pubished in History of Programming Languages, edited by Richard Wexelblat, Academic Press 1981. Two quotations: I spent the summer of 1958 at the IBM Information Research Department at the invitation of Nathaniel Rochester and chose differentiating algebraic expressions as a sample problem. It led to the following innovations beyond the FORTRAN List Processing Language: • • • • (c) To use functions as arguments, one needs a notation for functions, and it seemed natural to use the λ-notation of Church (1941). I didn't understand the rest of his book, so I wasn't tempted to try to implement his more general mechanism for defining functions. Church used higher-order functionals instead of using conditional expressions. Conditional expressions are much more readily implemented on computers. • • • • Logical completeness required that the notation used to express functions used as functional arguments be extended to provide for recursive functions, and the LABEL notation was invented by Nathaniel Rochester for that purpose. D. M. R. Park pointed out that LABEL was logically unnecessary since the result could be achieved using only λ — by a construction analogous to Church's Y-operator, albeit in a more complicated way.

Other key McCarthy publications:

Recursive Functions of Symbolic Expressions and their Computation by Machine (Part I). The original paper on LISP from CACM, April 1960. Part II, which never appeared, was to have had some Lisp programs for algebraic computation.

A Basis for a Mathematical Theory of Computation, first given in 1961, was published by North-Holland in 1963 in Computer Programming and Formal Systems, edited by P. Braffort and D. Hirschberg.

Towards a Mathematical Science of Computation, IFIPS 1962 extends the results of the previous paper. Perhaps the first mention and use of abstract syntax.

Correctness of a Compiler for Arithmetic Expressions with James Painter. May have been the first proof of correctness of a compiler. Abstract syntax and Lisp-style recursive definitions kept the paper short.

An HTML site concerning Lisp history can be found at:

http://www8.informatik.uni-erlangen.de/html/lisp-enter.html

�16

A Closing Thought from Robert Harper For me, I think it is important to stress the overwhelming influence of the λ-calculus among all other models of computation: • It codifies not only computation, but also the basic principles of human reason (natural deduction). • Moreover, it was born fully formed, and is directly and immediately relevant to this day, rather than something that collects dust on the shelf.

Admittedly Turing's model had the advantage of being explicitly psychologically motivated, but on the other hand Church focused on one of the greatest achievements of the human mind, the concept of a variable (= reasoning under hypotheses). Church saw that this was central, and time has born out the significance of his insight.

By contrast, no one cares one bit about the details of a Turing Machine; for, it fails to address the central issue of modularity (logical consequence), which is so important in programming and reasoning. And it does not extend to higher-order computation in anything like a natural or smooth way.

λ CONQUERS ALL!

Perhaps my good friend and colleague has spoken a

little too strongly here, as Turing Machines have

had many applications, say in Complexity Theory.

But the study of Programming Languages

does not seem to need them today.